"native lens"

I've been reading reviews regarding the Olympus OMD EM1 (and others) and when referring to AF speed, they say things like, "AF was fast with this Four Thirds lens, but is faster with a native lens." Here, does 'native' mean Micro Four Thirds mount lenses (any brand), or does it require a narrower interpretation, meaning Micro Four Thirds mount lenses built by Olympus?

What I really want to know is if I can expect Micro Four Thirds lenses (in general, any brand, & assuming similar quality) to be as fast at AF as Olympus brand Micro Four Thirds lenses? In other words is it recommended I stick with Olympus lenses in general (for use with the OMD-EM1)? Or are other brands expected to perform just as well with this camera, provided I stick with the Micro Four Thirds mount? Are there other significant factors besides AF speed to watch out for when choosing "native" and "non-native"?

I'm new to cameras with detachable lenses, trying to figure out which ones to purchase for the OMD EM1.